and
the chain rule of mutual information [1]}; the second
inequality follows from the fact that conditioning reduces differential entropy
[1]}; and the third inequality follows from the well-known
fact that Gaussian maximizes differential entropy for a given covariance matrix
[1]}. By (REF ),
\(|\mathbf {I}-\mathbf {D}^{-2}|^{\frac{1}{n}}\exp \left[-\frac{2}{n}I(\mathbf {Z};\mathbf {D}\mathbf {X}+\mathbf {Z})\right]& \le \; |\mathbf {N}^{-1}{\sf Cov}(\mathbf {Z}|\mathbf {D}\mathbf {X}+\mathbf {Z})(\mathbf {I}-\mathbf {D}^{-2})|^{\frac{1}{n}}\nonumber \\&\le \; \frac{1}{n} {\sf Tr}\left\lbrace \mathbf {N}^{-1} {\sf Cov}(\mathbf {Z}|\mathbf {D}\mathbf {X}+\mathbf {Z})(\mathbf {I}-\mathbf {D}^{-2})\right\rbrace  \) 
